Invariance and stability of fuzzy systems

An Overview of Fuzzy Control Theory

“…Kloeden [76], and research concerning the stability of a fuzzy system makes use of the concept of energy or controllability of the fuzzy system. See for instance [42] and [75]. The first articles specifically on stability of fuzzy controllers are due to C.V.…”

Discrete dynamical systems in L-topological spaces

“…Such a limit was calculated by the corresponding "membership functions" A n and A and therefore needed a "distance" between functions A n and A or something like "open neighborhoods" of fuzzy sets [3]. Moreover, discussion in [5] was actually aimed at its -cut. In this paper, the concept of limit sets is generalized to lattices in a different manner.…”

“…The concept of limit sets was considered in [5] for flows, where it was actually defined by a limit of a sequence of fuzzy sets A n to another fuzzy set A. Such a limit was calculated by the corresponding "membership functions" A n and A and therefore needed a "distance" between functions A n and A or something like "open neighborhoods" of fuzzy sets [3].…”

“…The rapid advances in the theories of fuzzy sets and fuzzy topology indicate a way to overcome the difficulty. Many efforts [4,5,14] have been made in this direction. In [4], a theory of dynamical fuzzy systems was proposed.…”

“…It was shown that a fuzzy system could be presented by a flow, describing the evolution of this fuzzy system in the space of fuzzy subsets of the state-space. Furthermore, in [5] Glas proposed an Invariance Principle for fuzzy systems, which generalized the LaSalle's Invariance Principle [16], and investigated three concepts of stability for a fuzzy system, which generalized the corresponding concepts in the known theory of Liapunov stability. Kloeden [14] defined a fuzzy dynamical system on an underlying complete, locally compact metric state space X in terms of a fuzzy attainability set mapping on X.…”